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Line 1: − + + + + + − An approximate value for '''π''' is 3.141592653... The exact value can never be expressed as a fraction or as a decimal number, regardless of how many digits are used (Lambert<ref>{{cite web+ + + − + − + − − The approximate value quoted above is accurate enough to calculate the circumference of a circle of the size of the Earth with an error of only about 1 millimetre. For rough approximations, the fraction 22/7 (= 3.142857...), correct to 2 decimal places, is sometimes used. A more accurate fractional approximation is 355 / 113 (= 3.14159292...), which is correct to 6 decimal places. − The Greek letter '''π''' is the sixteenth letter of the [[Greek alphabet]]. It was first used with its current meaning in 1706 by a Welsh mathematician William Jones <ref name="hist">{{cite web+ − + − + − + − − + − + − + − + − + − <big><math>\pi</math></big> Approximation Day is a similar holiday, celebrated on 22nd July (from the approximation 22/7). <ref>[http://www.usatoday.com/tech/science/mathscience/2007-03-14-pi-day_N.htm USA Today (3/14/2007) - Pi-day]</ref>+ − <big><math>\pi</math></big> is approximately:+ − + − + − + − + − + − − + − + Line 77:
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Change to "pi" in running text. That's how the dictionaries give it. Making usage consistent through the article
'''Pi''' or '''<big><math>\pi</math></big>''' or '''π''' is the name or symbol for an important mathematical constant, which appears in many branches of [[mathematics]]. '''π''' was originally defined in geometric terms as the ratio of the circumference of a circle to its diameter. A simple well-known formula using '''π''' gives the area of a [[circle]] as: '''π''' * radius * radius. <ref>{{cite web
'''Pi''' is an important mathematical constant. It is defined as the ratio of the circumference of a [[circle]] to its diameter. In formulas, pi is given as a Greek letter (<big><math>\pi</math></big>) while diameter expressed as "2r" (double the radius). So the relation is:
:<big><math>C=2 \pi r</math></big>
Pi may be used to determine the area of a circle:<ref>{{cite web
| url = http://www.worsleyschool.net/science/files/circle/area.html
| url = http://www.worsleyschool.net/science/files/circle/area.html
| title = The Circle Area Formula
| title = The Circle Area Formula
| accessdate = 2012-02-13}}</ref>
| accessdate = 2012-02-13}}</ref>
:<big><math>Area= \pi r^{2}</math></big>
The value of pi is approximately 3.14 or 22/7. The exact value cannot be expressed as a fraction or as a decimal number, regardless of how many digits are used.<ref>{{cite web
| url = http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Lambert.html
| url = http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Lambert.html
| title = Biography of Johann Heinrich Lambert
| title = Biography of Johann Heinrich Lambert
| accessdate = 2012-02-12}}</ref> proved this in 1761 by showing that '''π''' is an [[irrational number]], which means that it can't be expressed as a fraction; Lindeman<ref>{{cite web
| accessdate = 2012-02-12}}</ref> proved this in 1761 by showing that pi is an [[irrational number]], which means that it can't be expressed as a fraction. In 1882, Lindeman proved that pi is also a [[transcendental number]], which means that it can't be expressed as the solution to any simple equation).<ref>{{cite web
| url = http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Lindemann.html
| url = http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Lindemann.html
| title = Biography of Carl Louis Ferdinand von Lindemann
| title = Biography of Carl Louis Ferdinand von Lindemann
| accessdate = 2012-02-12}}</ref> went further in 1882 and proved that '''π''' is a [[transcendental number]] which means that it can't be expressed as the solution to any simple equation).
| accessdate = 2012-02-12}}</ref>
==History==
==History==
Pi is the sixteenth letter of the [[Greek alphabet]]. It was first used with its current meaning in 1706 by a Welsh mathematician William Jones.<ref name="hist">{{cite web
| url = http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Pi_through_the_ages.html
| url = http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Pi_through_the_ages.html
| title = A history or Pi
| title = A history or Pi
| accessdate = 2012-02-11}}</ref> because it is the first letter of the Greek word for [[perimeter]] (the circumference of a circle is its perimeter).
| accessdate = 2012-02-11}}</ref> because it is the first letter of the Greek word for [[perimeter]] (the circumference of a circle is its perimeter).
Mathematicians have worked for centuries to calculate <big><math>\pi</math></big> to more and more decimal places. To some extent, the progress of mathematics—or at least of computation—can be gauged by the progress in the number of digits to which <big><math>\pi</math></big> has been calculated.
Mathematicians have worked for centuries to calculate pi to more and more decimal places. To some extent, the progress of mathematics—or at least of computation—can be gauged by the progress in the number of digits to which pi has been calculated.
Some ancients expressed <big><math>\pi</math></big> by using fractional approximations. Papyrus of Ahmes, dated c. 1650 B.C., shows that ancient Egyptians had value 3 1/6 = 3.167). The Babylonian value from the same era was 3 1/8 = 3.125<ref>Boyer, A History of Mathematics, 2nd Edition</ref>. Both these values are accurate to within 1 percent. Note that the value 22/7 (3 1/7) is still used today.
Some ancients expressed pi by using fractional approximations. Papyrus of Ahmes, dated c. 1650 B.C., shows that ancient Egyptians had value 3 1/6 = 3.167). The Babylonian value from the same era was 3 1/8 = 3.125<ref>Boyer, ''A History of Mathematics'', 2nd edition</ref>. Both these values are accurate to within 1 percent. Note that the value 22/7 is still used today.
[[Archimedes]] of Syracuse (287-212 BC) carried out "the first theoretical calculation" of <big><math>\pi</math></big>.<ref>[http://veling.nl/anne/templars/Pi_through_the_ages.html Pi through the ages]</ref>
[[Archimedes]] of Syracuse (287-212 BC) carried out "the first theoretical calculation" of pi.<ref>[http://veling.nl/anne/templars/Pi_through_the_ages.html Pi through the ages]</ref> He said it was between 223/71 and 22/7. This is ten times better than the Egyptian and Babylonian values, and it is within 0.04 percent of the correct value.
He said it was between 223/71 and 22/7. This is ten times better than the Egyptian and Babylonian values: within 0.04% of <big><math>\pi</math></big>.
In 1873, Abraham Shanks spent twenty years calculating <big><math>\pi</math></big> to 707 places, but made a mistake in his calculation and only 527 of them were correct. When electronic computers were developed, <big><math>\pi</math></big> was soon calculated to tens of thousands, millions, and billions of places. As of 2002, the record is held by Yasumasa Kanada of Tokyo University at 1,241,100,000,000 digits.<ref>http://www.super-computing.org/pi_current.html</ref> That result was never printed out.
In 1873, Abraham Shanks spent twenty years calculating pi to 707 places. Due to an error in his calculations, only the first 527 digits were correct. When electronic computers were developed, pi was calculated to trillions of places. In 2014, "houkouonchi" calculated the first 13.3 trillion digits of pi.<ref>Yee, Alexander, "[http://www.numberworld.org/y-cruncher/ y-cruncher - A Multi-Threaded Pi-Program]"</ref> The result has not been printed out.
==Pi in mathematics==
==Pi in mathematics==
It's impossible to overestimate the importance of <big><math>\pi</math></big> (and [[e]]) for mathematics. Both values are intrinsically tied, e.g. by [[Euler's identity]]
It's impossible to overestimate the importance of pi (and [[e]]) for mathematics. These values are tied by [[Euler's identity]]:
<math> e^{\pi \imath} +1 = 0</math>.
:<math>e^{\pi \imath} +1 = 0</math>.
==Recreation==
==Recreation==
Memorizing <big><math>\pi</math></big> is a challenge that appeals to some people. Mnemonics have been devised. Counting the letters in each word of the phrase "Now I want a drink—alcoholic, of course" gives <big><math>\pi</math></big> to seven places (which is more than enough for all ordinary purposes). Numerous other mnemonics of this kind have been devised; in 1995, Michael Keith wrote one entitled [http://users.aol.com/s6sj7gt/mikerav.htm Near a Raven] which simultaneously parodies [[Edgar Allen Poe]]'s poem ''The Raven,'' while encoding <big><math>\pi</math></big> to 740 places.
Memorizing pi is a challenge that appeals to some people. Mnemonics have been devised. Counting the letters in each word of the phrase "Now I want a drink—alcoholic, of course" gives pi to seven places (which is more than enough for all ordinary purposes). Numerous other mnemonics of this kind have been devised; in 1995, Michael Keith wrote one entitled [http://users.aol.com/s6sj7gt/mikerav.htm Near a Raven] which simultaneously parodies [[Edgar Allen Poe]]'s poem ''The Raven,'' while encoding <big><math>\pi</math></big> to 740 places.
March 14 marks [[Pi Day]], a holiday on which the mathematical constant is celebrated. The date, 3/14, comes from the first three digits of <big><math>\pi</math></big>; some people begin their celebration at 1:59 pm, derived from the following three digits.
March 14 marks [[Pi Day]], a holiday on which the mathematical constant is celebrated. The date, 3/14, comes from the first three digits of pi. Some people begin their celebration at 1:59 pm, derived from the following three digits.
Pi Approximation Day is a similar holiday, celebrated on 22nd July (from the approximation 22/7).<ref>[http://www.usatoday.com/tech/science/mathscience/2007-03-14-pi-day_N.htm USA Today (3/14/2007) - Pi-day]</ref>
The value of pi is approximately:
{{quotebox|
{{quotebox|
3.14159265358979323846264338327950288419​7169399375​1058209749​4459230781​6406286208​9986280348​2534211706​7982148086​5132823066​4709384460​9550582231​7253594081​2848111745​0284102701​9385211055​5964462294​8954930381​9644288109​7566593344​6128475648​2337867831​6527120190​9145648566​92...}}
3.14159265358979323846264338327950288419​7169399375​1058209749​4459230781​6406286208​9986280348​2534211706​7982148086​5132823066​4709384460​9550582231​7253594081​2848111745​0284102701​9385211055​5964462294​8954930381​9644288109​7566593344​6128475648​2337867831​6527120190​9145648566​92...}}
== Greek Language Usage ==
== Greek language usage ==
This letter's name is pronounced the same as its equivalent Latin letter in English (P), and has the phonetic value, /p/.
This letter's name is pronounced the same as its equivalent Latin letter in English (p), and has the phonetic value, /p/.
==Does the Bible attempt to define Pi?==
==Does the Bible attempt to define pi?==
Virtually all serious students of the [[Bible]] say no. Still, critics frequently claim that the Bible contains an incorrect value for <big><math>\pi</math></big>, and the question is raised frequently enough to earn mention in the [[Skeptics Annotated Bible]].
Virtually all serious students of the [[Bible]] say no. Still, critics frequently claim that the Bible contains an incorrect value for pi, and the question is raised frequently enough to earn mention in the [[Skeptics Annotated Bible]].
The claim is based on a verse in the [[I Kings|first book of Kings]]:
The claim is based on a verse in the [[I Kings|first book of Kings]]:
{{bible quote|He made the Sea of cast metal, circular in shape, measuring ten [[cubit]]s from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.|book=1_Kings|chap=7|verses=23|version=NIV}}
{{bible quote|He made the Sea of cast metal, circular in shape, measuring ten [[cubit]]s from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.|book=1_Kings|chap=7|verses=23|version=NIV}} Thus, critics say, the Bible claims that the value of pi is three, and is therefore in error. However, there are a number of assumptions in this claim, any of which might invalidate the argument if wrong:
Thus, critics say, the Bible claims that the value of <big><math>\pi</math></big> is 3, and is therefore in error. However, there are a number of assumptions in this claim, any of which might invalidate the argument if wrong:
* That the tools and system of measurement available to the builders were sufficiently accurate to distinguish between 3 and pi.
* That the tools and system of measurement available to the builders were sufficiently accurate to distinguish between three and pi.
* That the phrase translated as "circular in shape" means perfectly circular, not simply "round" as an ellipse is round.
* That the phrase translated as "circular in shape" means perfectly circular, not simply "round" as an ellipse is round.
* That the Bible is trying to provide a value for <big><math>\pi</math></big>, and not merely describing the historical event of building this object.<ref name="math">{{cite web
* That the Bible is trying to provide a value for pi, and not merely describing the historical event of building this object.<ref name="math">{{cite web
| url = http://mathforum.org/library/drmath/view/52573.html
| url = http://mathforum.org/library/drmath/view/52573.html
| title = Discussion re rounding Pi
| title = Discussion re rounding Pi
{{reflist}}
{{reflist}}
==See Also==
==See also==
*[[Pi day]]
*[[Pi Day]]
*[http://yacas.sourceforge.net/Algochapter5.html#c5s5 Calculation of pi with computers]
*[http://yacas.sourceforge.net/Algochapter5.html#c5s5 Calculation of pi with computers]
[[Category:Mathematics]]
[[Category:Mathematics]]